A079510 → basis A013611

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(1 + 4\,x + n + 4\,n x\right)\,P_{n-2}(x) + \left(1 + 4\,x + n + 4\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	9	60	96													165
3	44	448	1472	1536												3500
4	265	3600	18000	39040	30720											91625
5	1854	31704	213984	710784	1158144	737280										2853750
6	14833	306460	2612400	11744320	29317120	38449152	20643840									103088125
7	133496	3239296	33432320	190085120	642365440	1288699904	1419116544	660602880								4237675000
8	1334961	37251648	452049696	3114074880	13311002880	36124913664	60735209472	57776799744	23781703680							195334340625
9	14684570	463647720	6474457920	52460880640	271700083200	932270868480	2118087802880	3070547066880	2575313141760	951268147200						9978600781250
10	176214841	6214752412	98231155968	916090845824	5580419989760	23192841391104	66576467550208	130279775076352	166236293431296	124824122818560	41855798476800					559566431703125
11	2290792932	89303973312	1577030572992	16648490184192	116715923745792	570403110617088	1982340052697088	4897542522667008	8426554741751808	9612261807095808	6539488348078080	2009078326886400				34172701949062500
12	32071101049	1370027838608	26745376214128	315453163296128	2503210682123008	14076157250355200	57502904267640832	171905348923523072	373153042940035072	573407250918408192	591885942260760576	368347913870376960	104472072998092800			2257597444749765625
13	481066515734	22354911429816	478236276487200	6235477928803200	55276665077061120	351767734982264832	1653000800699179008	5806414162185584640	15243076288531660800	29530730648337121280	41027048741151965184	38696822004856651776	22202600585020047360	5850436087893196800		160423910268917968750
14	7697064251745	386662346033340	8998221146625360	128562865543569600	1259754001913721600	8954450929040348160	47608189108459499520	192305014964645806080	592942610638621900800	1388535828377134694400	2430532294206522654720	3083113629785668976640	2678516284993893826560	1426180900799486361600	351026165273591808000	12201113078525080078125

Technical details

Node id: n-f209d0e79422f087 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-28fe48d946165a82

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        2,         8;
  2:        9,        60,         96;
  3:       44,       448,       1472,        1536;
  4:      265,      3600,      18000,       39040,        30720;
  5:     1854,     31704,     213984,      710784,      1158144,       737280;
  6:    14833,    306460,    2612400,    11744320,     29317120,     38449152,      20643840;
  7:   133496,   3239296,   33432320,   190085120,    642365440,   1288699904,    1419116544,     660602880;
  8:  1334961,  37251648,  452049696,  3114074880,  13311002880,  36124913664,   60735209472,   57776799744,   23781703680;
  9: 14684570, 463647720, 6474457920, 52460880640, 271700083200, 932270868480, 2118087802880, 3070547066880, 2575313141760, 951268147200;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(4*n+4)*x)*P(n-2,x) + (n+1+(4*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (4*n + 4)*x)*P[n - 2, x] + (n + 1 + (4*n + 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (4*n + 4)*x)*v[i - 2] + (n + 1 + (4*n + 4)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows; related to number of preorders.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003